Contractions 

       Fractal is a set of transformations. We can also use the term contraction to describe these transformations. Each transformation results in a set of shapes that are smaller than the previous shape, hence the name contraction. To put it more simply, a contraction is a scaling of an element of a set by a factor 1/r, where r is an integer.

        A set of contractions on the same set is called a contraction mapping. A set of contraction mappings applied in succession is an IFS. 

Dimension

       Dimension is a term used to measure the size of a set. Usually, this set will be an image, but not always. It helps to think of images or objects when thinking about dimension. We are all familiar with objects that are one-dimensional (a line segment), two-dimensional (a square), and three-dimensional (a cube).

       We can also say that these objects have dimension 1, 2, and 3, respectively. Fractals have dimension, also. But the value of their dimension does not necessarily need to be an integer. This fact is what gives fractals many of their unique properties; for example, the Sierpinski triangle has an area of 0.

     There are two main types of dimension that we measure: box dimension and topological dimension. We now explore the two. 

Box Dimension

     Suppose we have a number of boxes, all with the same side length. We denote this length by r. Now suppose we have an object in R^n and we want to cover this object with these boxes. Let (1/r)^n  denote the number of such boxes it takes to cover this object. These boxes have area r^n , and this they were scaled by a factor of (1/r)^n .

Now, if we took a simple square of length s and covered it with boxes of area r^n , we could determine (1/r)^n as follows:  

S^2=N(r)*r^n

N(r)=S^2/r^n

N(r)=S^2(1/r)^n

 

Since S^2 is a constant, we can denote it by C, thus giving us:

N(r)=C(1/r)n

Solving for n yields:

n=(lnN(r)-lnC) /ln(1/r)

n is the dimension of our object. Since C is a constant, we can ignore it for our purposes. If we take the limit of this formula as r approaches 0, we get the formula for box dimension:

Using the Sierpinski triangle as an example, we have N(r) = 3 (three smaller triangles created from one large one) and r = 1/2(each triangle is scaled by a factor of 1/2). Plugging those numbers in, we get

DB(S)=ln3/ln2=1.5850

which is the box dimension for the Sierpinski triangle. 

Topological Dimension

            Topological dimension is more difficult to explain and compute. A few common definitions are used, all of which are analogous to one another. The definition we will use deals with line segments. If a set S cannot be restated as the union of two or more sets, and this set does not contain any line segments, then the topological dimension of S is 0. If a set S is a union of two or more sets of topological dimension k, where k

           The set of rational numbers and the set of irrational numbers do not contain any line segments, and thus both have topological dimension 0.

         The union of these two sets is the set of real numbers, which thus has topological dimension 1.

        Often, the topological dimension cannot be computed exactly. For simple objects, like lines, polygons, and polyhedrons, the topological dimension is equal to the box dimension.

DB(S)=Dt(S)

        A fractal is a set for which DB(S)>Dt(S)  . This is the formal mathematical definition. Mathematica can compute the fractal dimension (box dimension) of a set.

Some Theory and Definitions

          The purpose of this section is to provide a basic theoretical background on fractals for those who are interested. The more technical terms used here will not be used anywhere else on the site.It is not necessary to have an understanding of this sction in order to understand the rest of the site.In this section, we will define a fractal using some other terms that are often used when dealing with fractals mathematically. We will also briefly go into chaos theory, a banch of mathematics that is closely related to fractal geometry.